Theorem dfrnf 4788

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	⊢ Ⅎ𝑥𝐴
dfrnf.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
dfrnf	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 4735	. 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2		nfcv 2282	. . . . 5 ⊢ Ⅎ𝑥𝑣
3		dfrnf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2282	. . . . 5 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 3982	. . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤
6		nfv 1509	. . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤
7		breq1 3940	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤))
8	5, 6, 7	cbvex 1730	. . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
9	8	abbii 2256	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10		nfcv 2282	. . . . 5 ⊢ Ⅎ𝑦𝑥
11		dfrnf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
12		nfcv 2282	. . . . 5 ⊢ Ⅎ𝑦𝑤
13	10, 11, 12	nfbr 3982	. . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤
14	13	nfex 1617	. . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤
15		nfv 1509	. . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦
16		breq2 3941	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1798	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2264	. 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2165	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1332  ∃wex 1469  {cab 2126  Ⅎwnfc 2269   class class class wbr 3937  ran crn 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  rnopab  4794